Renormings in Banach Spaces

“I should say that this very interesting book is exactly what it claims to be: a toolbox, which is designed to be as useful as possible to all mathematicians who wish to understand this exciting and deep field of research. Please read, and enjoy!” (Gilles Godefroy, zbMATH 1508.46001, 2023)

• 1 Norms, Normed spaces, Banach spaces.- 2 Some basic definitions and tools.- 3 Equivalent norms.- 4 Basic differentiability in Banach spaces.- 5 Basic convexity.- 6 Some structural properties of Banach spaces.- 7 The use of Smulyan’s tests.- 8 Asplund averaging I.- 9 Tools for renorming.- 10 Renorming of nonseparable Banach spaces  .- 11 Examples on C1-smoothness.- 12 Examples on Rotundity.- 13 Nonlinear Transfer Techniques.- 14 Lipschitz functions.- 15 Spaces isomorphic to Hilbert spaces.- 16 Superreflexive spaces.- 17 The Kingdom of Tsirelson’s space.- 18 The L(infinity) spaces.- 19 Higher order smoothness.- 20 James boundaries .- 21 RNP property.- 22 SSD spaces.- 23 Norms with MIP.- 24 Nicely smooth Banach spaces.- 25 Weak Hadamard differentiability.- 26 Fabian’s farm of Lipschitz Asplund spaces.- 27 Fonf’s land of spaces isomorphic to polyhedral spaces.- 28 Hájek’s garden of nice functions on c0(gamma).- 29 Kottman-type results.- 30 3-space properties.- 31 Polynomials.- 32 Miscellaneous applications.- 33 Miscellaneous topics.- 34 More on WCG spaces and their relatives.- 35 Valdivia compacta.- 36 Renorming classical spaces.- 37 Symmetric norms.- 38 A concise list of coordinates of the relationship.- 39 Some easily formulated open questions on construction of norms in separable spaces.- Index of figures.